Calculating Expected Value: Harlene's Dice Game

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Calculating Expected Value: Harlene's Dice Game

Hey guys! Let's dive into a fun probability problem involving Harlene and her dice. The core of this problem revolves around calculating the expected value of a game where Harlene rolls two number cubes. This concept is super useful in all sorts of areas, from financial investments to game strategies. So, let's break it down step by step and make sure we completely understand how to solve it.

The Setup: Harlene's Dice Roll Game

Here's the deal: Harlene rolls two standard six-sided dice. If she rolls a sum of 8 or 12, she scores 9 points. But, if she rolls anything else, she loses 2 points. Our goal is to figure out the expected value of her score for a single roll. Understanding expected value helps us predict, on average, what Harlene's score will be over many, many rolls.

To solve this, we will go through these steps:

  1. Identify Outcomes: First, we'll determine all the possible sums she can roll with two dice.
  2. Calculate Probabilities: Next, we'll figure out the probability of each outcome, especially the sums of 8 and 12.
  3. Determine Points: Then, we will figure out the points Harlene gets for each outcome.
  4. Calculate Expected Value: Finally, we'll use a formula to calculate the expected value. This involves multiplying the value of each outcome by its probability and summing these products.

By following these steps, we'll be able to determine what Harlene can expect to score on average.

Step 1: Identifying Possible Outcomes

Alright, let's start by listing all the possible sums Harlene can get when she rolls two six-sided dice. The minimum sum she can roll is 2 (1 + 1), and the maximum is 12 (6 + 6). Here's a list of all the possible sums:

  • 2
  • 3
  • 4
  • 5
  • 6
  • 7
  • 8
  • 9
  • 10
  • 11
  • 12

These are all the potential outcomes we need to consider.

Step 2: Calculating Probabilities for each Outcome

Now, let's calculate the probabilities of Harlene rolling each of these sums. This involves figuring out how many combinations of dice rolls result in each sum and dividing that by the total number of possible outcomes. Since each die has six sides, there are 6 * 6 = 36 total possible outcomes.

Here's a breakdown of the probabilities for the sums we care about (8 and 12):

  • Sum of 8: There are 5 ways to get a sum of 8: (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2). So, the probability of rolling an 8 is 5/36.
  • Sum of 12: There is only 1 way to get a sum of 12: (6, 6). The probability of rolling a 12 is 1/36.
  • Probability of NOT 8 or 12: To get this, we add the probability of 8 and 12 and subtract the result from 1: 1 - (5/36 + 1/36) = 1 - 6/36 = 30/36

These probability calculations are crucial because they form the foundation for determining the expected value.

Step 3: Determining Points for each Outcome

Next, let's define the points Harlene receives for each outcome:

  • Sum of 8 or 12: She gets 9 points.
  • Any other sum: She loses 2 points (which means she gets -2 points).

These point values are what we'll use to calculate the expected value.

Step 4: Calculating the Expected Value

Finally, we're ready to calculate the expected value. The formula for expected value is:

Expected Value = (Probability of Outcome 1 * Value of Outcome 1) + (Probability of Outcome 2 * Value of Outcome 2) + ...

In our case, we have two outcomes to consider:

  1. Rolling an 8 or 12:
    • Probability: 6/36
    • Value: 9 points
  2. Rolling any other sum (not 8 or 12):
    • Probability: 30/36
    • Value: -2 points

Now we plug these values into the formula:

Expected Value = (6/36 * 9) + (30/36 * -2)

Let's calculate:

Expected Value = (54/36) + (-60/36)

Expected Value = -6/36

Expected Value = -1/6

Therefore, the expected value of Harlene's score for one roll is -1/6.

Conclusion: Understanding Expected Value in Dice Games

So, the expected value of Harlene's dice game is -1/6. This means that, on average, Harlene will lose 1/6 of a point per roll over many rolls. This negative value indicates that it's not a favorable game for her in the long run. By understanding how to calculate expected value, we can analyze the potential outcomes of games and make informed decisions about whether to play or not.

In this case, Harlene is expected to lose points, highlighting the importance of understanding the probabilities and potential outcomes involved in games of chance. The calculation itself is straightforward: identify the outcomes, determine their probabilities, assign the values, and then apply the expected value formula. This provides a clear framework for analyzing the fairness and profitability of different games. The concept of expected value isn't just limited to dice games. It's a fundamental concept that can be applied to all sorts of situations involving uncertainty, from financial investments to everyday decision-making.

So, whether you're analyzing a dice game or considering an investment, the ability to calculate expected value gives you a powerful tool to predict outcomes and make better decisions. Remember, the key is to understand the probabilities and the potential rewards and losses associated with each outcome.